Lecture 5

# Lecture 5 - Nash Equilibrium Consider the battle of the...

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Nash Equilibrium Consider the battle of the sexes Man\Woman opera ballet opera 1,4 0,0 ballet 0,0 4,1 In this game, there is no dominant strategy. But suppose W is playing opera. Then, the best thing M can do is to play opera, too. Thus opera is a best-response for M against opera. Similarly, opera is a best-response for Wagainst opera. Thus, at (opera, opera), neither party wants to take a different action. This is a Nash Equilibrium. More formally: Definition 12 For any player i, a strategy s BR i is a best response to s −i if and only if u i (s BR i , s −i ) ≥ u i (s i , s −i ), s i S i This definition is identical to that of a dominant strategy except that it is not for all s −i S −i but for a specific strategy s −i . If it were true for all s −i , then S BR i would also be a dominant strategy, which is a stronger requirement than being a best response against some strategy s −i . Definition 13 A strategy profile (s NE 1 , . ..s NE N ) is a Nash Equilibrium if and only if s NE i is a best-response to s NE −i = (s NE 1 , . ..s NE i−1 , s NE i+1 , . ..s NE N ) for each i. That is, for all i, we have that u i (s NE i , s NE −i

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Lecture 5 - Nash Equilibrium Consider the battle of the...

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